de

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Beliebt Rechnen >

integral of 1/(x^2(x^2-4))

Lösung

\int \frac{1}{x ^{2} ( x ^{2} - 4 )} d x

Lösung

\frac{1}{4 x} - \frac{1}{16} ln ∣ x + 2 ∣ + \frac{1}{16} ln ∣ x - 2 ∣ + C

Schritte zur Lösung

\int \frac{1}{x ^{2} ( x ^{2} - 4 )} d x

Ermittle den Partialbruch von

\frac{1}{x ^{2} ( x ^{2} - 4 )} : - \frac{1}{4 x ^{2}} - \frac{1}{16 ( x + 2 )} + \frac{1}{16 ( x - 2 )}

= \int - \frac{1}{4 x ^{2}} - \frac{1}{16 ( x + 2 )} + \frac{1}{16 ( x - 2 )} d x

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - \int \frac{1}{4 x ^{2}} d x - \int \frac{1}{16 ( x + 2 )} d x + \int \frac{1}{16 ( x - 2 )} d x

\int \frac{1}{4 x ^{2}} d x = - \frac{1}{4 x}

\int \frac{1}{16 ( x + 2 )} d x = \frac{1}{16} ln ∣ x + 2 ∣

\int \frac{1}{16 ( x - 2 )} d x = \frac{1}{16} ln ∣ x - 2 ∣

= - (- \frac{1}{4 x}) - \frac{1}{16} ln ∣ x + 2 ∣ + \frac{1}{16} ln ∣ x - 2 ∣

Vereinfache

= \frac{1}{4 x} - \frac{1}{16} ln ∣ x + 2 ∣ + \frac{1}{16} ln ∣ x - 2 ∣

Füge eine Konstante zur Lösung hinzu

= \frac{1}{4 x} - \frac{1}{16} ln ∣ x + 2 ∣ + \frac{1}{16} ln ∣ x - 2 ∣ + C

Graph

Beliebte Beispiele

derivative of sin(|x|)

\frac{d}{d x} (sin (∣ x ∣))

integral of ((x^4-2x^3+1))/(x^2)

\int \frac{( x ^{4} - 2 x ^{3} + 1 )}{x ^{2}} d x

integral of (14x+1)sqrt(7x^2+x)

\int (14 x + 1) 7 x^{2} + x d x

x^2(dy)/(dx)+x(x+2)y=e^x

x^{2} \frac{d y}{d x} + x (x + 2) y = e^{x}

integral of 7/((x-3)(x+4))

\int \frac{7}{( x - 3 ) ( x + 4 )} d x